Finite Group Databasebeta

← #11737910

Subgroups — #11737910

Order 896.

326 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90412C211yesmaximal, characteristic
2#90261C211yesmaximal, characteristic
3C7×#462C211yesmaximal, characteristic
4#89956C212yesmaximal
5#89954C212yesmaximal
6#89853C211yesmaximal, characteristic
7C4×#1144C211yesmaximal, characteristic
8#89623C211yesmaximal, characteristic
9#89603C211yesmaximal, characteristic
10#89571C212yesmaximal
11#89569C212yesmaximal
12C2×C7×Q16C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15Q8×Dic7C2212yes—
16Q8×Dic7C2212yes—
17#6507C2212yes—
18C2×#1144C2211yescharacteristic
19#6453—21no—
20#6453—22no—
21#6453C2212yes—
22#6452—22no—
23#6450—21no—
24#6432—21no—
25#6432C2211yescharacteristic
26C4×Dic14—21no—
27C4×Dic14C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×C4⋊C8C2211yescharacteristic
30C7×#103C2212yes—
31#6409C2212yes—
32#6409C2211yescharacteristic
33#6409C2212yes—
34#6391C2212yes—
35#6390C2212yes—
36#6389C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2211yescharacteristic
39#6381C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3374—71nomaximal, Sylow
42C2×C7×Q8C2312yes—
43C2×C7×Q8C2311yescharacteristic
44C2×Dic14—21no—
45C2×Dic14D411yescharacteristic
46C2×Dic14C2311yescharacteristic
47C7×Q16—24no—
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4—21no—
52C7×C4⋊C4C2311yescharacteristic
53C42×C7—21no—
54C42×C7C2311yescharacteristic
55#1144—22no—
56C4⋊Dic7—21no—
57C4⋊Dic7—22no—
58C4⋊Dic7—22no—
59C4⋊Dic7—22no—
60C4⋊Dic7D411yescharacteristic
61C4⋊Dic7C2312yes—
62C4⋊Dic7C2311yescharacteristic
63Dic7⋊C4—21no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4D412yes—
68C4×Dic7—21no—
69C4×Dic7—22no—
70C4×Dic7—22no—
71C4×Dic7C2312yes—
72C2×C7⋊C8C2311yescharacteristic
73C2×C7⋊C8C2311yescharacteristic
74#557—71no—
75#557—71no—
76#491—72no—
77#487—71no—
78#479—72no—
79#475—71no—
80#462—72no—
81#462D711yescharacteristic
82#446—71no—
83#441—72no—
84C4×Q16—71no—
85C7×Q8—21no—
86C7×Q8—22no—
87C7×Q8—22no—
88C7×Q8#4112yes—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2411yescharacteristic
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7—22no—
98C2×Dic7C2×D412yes—
99C2×Dic7C2×D412yes—
100Dic14—22no—
101Dic14—21no—
102Dic14—21no—
103Dic14—42no—
104C7×C8—22no—
105C7⋊C8—21no—
106C7⋊C8—21no—
107C2×Q16—71no—
108C2×Q16C2×D712yes—
109C4⋊Q8—72no—
110C4⋊Q8—71no—
111C4⋊Q8—141no—
112C4⋊Q8C2×D711yescharacteristic
113#125—72no—
114#125—142no—
115#125—142no—
116#125—141no—
117C4×Q8—71no—
118C4×Q8—72no—
119C4×Q8—72no—
120C4×Q8—71no—
121C4×Q8—141no—
122C4×Q8—141no—
123C4×Q8C2×D711yescharacteristic
124#107—71no—
125#106—72no—
126C4⋊C8—72no—
127C4⋊C8C2×D711yescharacteristic
128#103—71no—
129#103—72no—
130#103—72no—
131#103—72no—
132#103—71no—
133#103C2×D712yes—
134#97—72no—
135C4×C8—71no—
136C22×C7C22×D411yescharacteristic
137C4×C7—22no—
138C4×C7—21no—
139C4×C7—42no—
140C4×C7—41no—
141C4×C7#13712yes—
142C4×C7C2×#4111yescharacteristic
143C4×C72−1+411yescharacteristic
144Dic7—22no—
145Dic7—22no—
146Dic7—41no—
147Dic7—42no—
148Dic7—41no—
149C2×Q8—71no—
150C2×Q8—71no—
151C2×Q8—141no—
152C2×Q8C22×D712yes—
153C2×Q8C22×D711yescharacteristic
154Q16—24no—
155Q16—142no—
156C2×C8—71no—
157C2×C8—71no—
158C2×C8C22×D712yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—72no—
163C4⋊C4—71no—
164C4⋊C4—72no—
165C4⋊C4—71no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—141no—
173C4⋊C4—141no—
174C4⋊C4C22×D711yescharacteristic
175C42—21no—
176C42—72no—
177C42—142no—
178C42—142no—
179C42—141no—
180C42C22×D711yescharacteristic
181C2×C7#54811yescharacteristic
182C2×C7C2×#13711yescharacteristic
183C2×C7#57811yescharacteristic
184Q8—21no—
185Q8—22no—
186Q8—22no—
187Q8—142no—
188Q8—141no—
189Q8—141no—
190Q8—282no—
191Q8#115912yes—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—72no—
196C2×C4—72no—
197C2×C4—141no—
198C2×C4—142no—
199C2×C4—141no—
200C2×C4D4×D711yescharacteristic
201C2×C4D4×D711yescharacteristic
202C2×C4C23×D711yescharacteristic
203C8—22no—
204C8—141no—
205C8—141no—
206C7#337411yescharacteristic, complemented, Sylow
207C22C2×D4×D711yescharacteristic
208C4—21no—
209C4—22no—
210C4—41no—
211C4—42no—
212C4—142no—
213C4—142no—
214C4—282no—
215C4—281no—
216C4—281no—
217C4#648112yes—
218C4C2×#115911yescharacteristic
219C4#655411yescharacteristic
220C2#9028911yescharacteristic
221C2C2×#648111yescharacteristic
222C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socle, prime radicalC22×C7C22×D4
hypercenter#462D7
hypocenterC7#3374
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
206. C7#3374semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#904122
#902612
C7×#4622
#899562
#899542
#898532
C4×#11442
#896232
#896032
#895712
#895692
#33747